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There are two equivalent definitions for birings. Let CRing\mathsf{CRing} denote the category of commutative unital rings and ring homomorphisms.

Definition 1. A biring is a coring object in (CRing,)(\mathsf{CRing}, \otimes).

Definition 2. A biring is a ring AA equipped with a lift of the functor CRing(A,):CRingSet\mathsf{CRing}(A, -) \colon \mathsf{CRing} \to \mathsf{Set} through the forgetful functor to a functor CRingCRing\mathsf{CRing} \to \mathsf{CRing}.


Birings were introduced in this paper: